A070243 a(n) = Card{ k, phi(k) <= n }.

2, 5, 5, 9, 9, 13, 13, 18, 18, 20, 20, 26, 26, 26, 26, 32, 32, 36, 36, 41, 41, 43, 43, 53, 53, 53, 53, 55, 55, 57, 57, 64, 64, 64, 64, 72, 72, 72, 72, 81, 81, 85, 85, 88, 88, 90, 90, 101, 101, 101, 101, 103, 103, 105, 105, 108, 108, 110, 110, 119, 119, 119, 119, 127, 127

Offset: 1

Author

Benoit Cloitre, May 11 2002

Keywords

• easy
• nonn

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-118.
• Gérald Tenenbaum and Jie Wu, Exercices corrigés de théorie analytique et probabiliste des nombres, Collection SMF, Cours spécialisés, Numero 2, pp. 78-79.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• Matteo Caorsi and Sergio Cecotti, Geometric classification of 4d N=2 SCFTs, Journal of High Energy Physics, Vol. 2018 (2018), Article number 138; arXiv preprint, arXiv:1801.04542 [hep-th], 2018.

Crossrefs

Partial sums of A014197.

Cf. A000010, A082695.

Programs

PARI

for(n=1,150,print1(sum(i=1,100*n,if(sign(eulerphi(i)-n)+1,0,1)+if((eulerphi(i)-n),0,1)),","))

PARI

list(nmax) = my(s = 0); for(n = 1, nmax, s += invphiNum(n); print1(s, ", ")); \\ Amiram Eldar, Dec 23 2024, using Max Alekseyev's invphi.gp

Formula

Limit_{n->oo} a(n)/n = zeta(2)*zeta(3)/zeta(6) = 1.943596436820759205057... = A082695.

From Benoit Cloitre, Apr 12 2003: (Start)

a(n) = Sum_{k=1..n} A014197(k).

a(n) = (zeta(2)*zeta(3)/zeta(6))*n + O(n*exp(-c*sqrt(log(n)))) for a suitable constant c > 0. (End)